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    Area of Science:

    • Control Theory
    • Game Theory
    • Optimization

    Background:

    • Leader-follower games with time delays present significant challenges due to noncausality in strategy design.
    • Existing research on delay-free leader-follower games is extensive, but the introduction of time delays complicates solutions.

    Purpose of the Study:

    • To develop a robust methodology for solving difference and differential leader-follower games with time delays.
    • To establish conditions for the existence and uniqueness of Stackelberg strategies in delayed game systems.

    Main Methods:

    • Introduction of novel co-states to represent future control information and new states for past effects.
    • Development of sufficient and necessary solvability conditions for the difference game.
    • Derivation of explicit open-loop Stackelberg strategies using decoupled and symmetric Riccati equations.

    Main Results:

    • A unique open-loop Stackelberg strategy is guaranteed to exist for the difference game under the derived conditions.
    • Explicit solutions for the open-loop strategies are provided in terms of Riccati equations.
    • A unique strategy is also presented for continuous-time systems.

    Conclusions:

    • The proposed techniques effectively overcome the noncausality issue in delayed leader-follower games.
    • The study provides a complete framework for analyzing and solving both discrete and continuous time delayed Stackelberg games.
    • The findings contribute to advancing the theory and application of game theory in systems with time delays.